Generalizations of Efron's theorem

TL;DR

This paper extends Efron's theorem by proving two new versions that accommodate broader classes of functions and stronger monotonicity conditions, enhancing its applicability in probability and analysis.

Contribution

It introduces two novel generalizations of Efron's theorem, expanding its scope to functions in the $PF_n$ class and under stronger monotonicity assumptions.

Findings

Generalized Efron's theorem for $PF_n$ functions.

Extended theorems under stronger monotonicity conditions.

Provided a more general result for the second generalization.

Abstract

In this article, we prove two new versions of a theorem proven by Efron in [Efr65]. Efron's theorem says that if a function $\phi : \mathbb{R}^2 \rightarrow \mathbb{R}$ is non-decreasing in each argument then we have that the function $s \mapsto \mathbb{E}[\phi(X,Y)|X+Y=s]$ is non-decreasing. We name restricted Efron's theorem a version of Efron's theorem where $\phi : \mathbb{R} \rightarrow \mathbb{R}$ only depends on one variable. $PF_n$ is the class of functions such as $\forall a_1 \leq ... \leq a_n, b_1 \leq ... \leq b_n, \det(f(a_i-b_j))_{1 \leq i,j \leq n} \geq 0.$ The first version generalizes the restricted Efron's theorem for random variables in the $PF_n$ class. The second one considers the non-restricted Efron's theorem with a stronger monotonicity assumption. In the last part, we give a more general result of the second generalization of Efron's theorem.